In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, given a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''X'' with an associated
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of
real bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
s,
definite quadratic forms and
isotropic quadratic form
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector ...
s are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.
A quadratic space which has a null vector is called a
pseudo-Euclidean space.
A pseudo-Euclidean vector space may be decomposed (non-uniquely) into
orthogonal subspaces ''A'' and ''B'', , where ''q'' is positive-definite on ''A'' and negative-definite on ''B''. The null cone, or isotropic cone, of ''X'' consists of the union of balanced spheres:
The null cone is also the union of the
isotropic lines through the origin.
Examples
The
light-like
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
vectors of
Minkowski space are null vectors.
The four
linearly independent biquaternions , , , and are null vectors and can serve as a
basis for the subspace used to represent
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. Null vectors are also used in the
Newman–Penrose formalism approach to spacetime manifolds.
[Patrick Dolan (1968]
A Singularity-free solution of the Maxwell-Einstein Equations
Communications in Mathematical Physics
''Communications in Mathematical Physics'' is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, s ...
9(2):161–8, especially 166, link from Project Euclid
A
composition algebra ''splits'' when it has a null vector; otherwise it is a
division algebra.
In the
Verma module of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
there are null vectors.
References
*
*
* {{cite book , last = Neville , first = E. H. (Eric Harold) , author-link =Eric Harold Neville , title =Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions , publisher =
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pr ...
, date = 1922 , pag
204 url =https://archive.org/details/prolegomenatoana00nevi
Linear algebra
Quadratic forms